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 elastic of a plane of the material which in the natural state was horizontal; all the boundaries of such a portion become parabolas of small curvature, which is variable along the length hJof the beam, and the particular effect under consideration is the change of the transverse horizontal linear elements from straight lines such as HK to parabolas such as H'K' (Fig. 5); the lines HL and 1 IvM are parallel to the central ^4 M line, and the figure is drawn for a plane above the neutral plane. Fig. 5. ,i character , the cross-section not the an ir elhpse the of When the strain is the same,is but curves are only approximately parabolic. Jrain corresponding to U is a distortion

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735 and makes an angle s0 with the horizontal (see Fig. 8); it is, however, improbable that this condition is exactly realized in practice. In the application of the theory to the experimental determination of Young’s modulus, the small angle which the central line at the support makes with, the horizontal is an unknown quantity, to be eliminated, by observation of the deflexion at two or more points. 8. We may suppose the displacement in a bent beam to be. produced by the. following operations : (1) the central me is deflected into its curved form, (2) the cross-sections are rotated about axes through their centroids at right angles to the. plane of flexure so as to make angles equal to 2~ + with the central line, (3) each cross-section is distorted in its own plane in such a way that the appropriate variable anticlastic curvature is produced, (4) the crosssections are further distorted into curved surfaces. The Wme curved Uni ^ ^ ^ ““ contour lines of Fig. 7 show the disturbance from the central tangent plane,, not from the original vertical plane. which cut the longi9’ Practical Application of Saint- Venant’s Theory.— tudinal filaments The theory above described is exact provided the forces obliquely, and thus applied to the loaded end, which have W for resultant the cross-sections do are distributed over the ternot remain plane, minal section in a particular but become curved way, not.likely to be realized surfaces, and the in practice and the applitangent plane to any cation to practical problems one of these surfaces depends on a principle due at the centroid cuts to Saint-Venant, to the effect the central line obthat, except for comparaliquely (Fig. 6). The tively small portions of the angle between these beam near to the loaded and tangent planes and fixed ends, the resultant Fig. 6. the central line is the same at all points, of the line; and, denoting it by only is effective, and its mode of distribution does 4 + 5o> the value of s0 is expressible as not seriously affect the inshearing stress at centroid Fig. 8. ternal strain and stress. In rigidity of material ’ fact, the actual stress is that due to forces with the required and it thus depends on the shape of the cross-section : for resultant distributed in the manner contemplated in the the elliptic section of § 4 its value is theory, superposed upon that due to a certain distribution 4W 2a2(l + o-) + 52 _ of forces on each terminal section which, if applied to a 3a2 + b'2 ’ rigid body, would keep it in equilibrium; according to for a circle with o- = ±, this becomes 7W/2E7ra2. The Saint-Yenant’s principle, the stresses and strains due to vertical filament through the centroid of any cross-section such distributions of force are unimportant except near becomes a cubical parabola, as shown in Fig. 6, and the the ends. For this principle to be exactly applicable it is contour, lines of the curved surface into which any cross- necessary that the length of the beam should be very section is distorted are shown in Fig. 7 for a circular section. great compared with any linear dimension of its cross7. The deflexion of the beam is determined from the section ; for the practical application it is sufficient that the length should be about ten times the greatest equation diameter. curvature of central line = bending moment ^-flexural rigidity, 10. The theoretical determination of the stress in a and the special con- bent beam under conditions as to load and support other ditions at the sup- than those considered in §§ 2-8 is attended by difficulties ported end; there which have not yet been surmounted, but the equation for is no alteration the deflexion of this statement curvature of central line =: bending moment-r flexural rigidity on account of the shears. As regards is sufficiently exact whenever the length is a considerthe special condi- able. multiple of the greatest diameter of the crosstion at an end section. This result is indicated by the theories of inwhich is encastree, definitely thin wires developed by Kirchhoff and Boussinesq, or built-in, Saint- and has been confirmed by special researches made by Venant proposed to Pochhammer and Pearson. The equation for the deflexion assume that the above written is the basis of the treatment of continuous central tangent beams resting on three or more supports and carrying plane of the cross- distributed loads. The calculation of the bending moment section at the end can be replaced by a method of graphical construcis vertical; with tion, due to Mohr, and depending on the two- following Fig. 7. this assumption the theorems :— tangent to the central line at the end is inclined downwards (I.) The curve of the central line of each span of a